The phrase "a parallel relationship with" is correct and usable in written English.
It can be used to describe a relationship that is similar or comparable to another relationship in some way.
Example: "The study found a parallel relationship with the economic growth and the increase in employment rates."
Alternatives: "a comparable relationship to" or "a similar relationship with".
He was regularly having sex with her at the age of 12. His brother was having a parallel relationship with her, although she did not see him as often.
I thought that after I divorced, I would find a parallel relationship with a special guy.
Concerning severe PUD, Nomura et al. [ 31] found that the incidence of hematemesis due to GU over the year showed an inverse temporal relationship to temperature and relative humidity and a parallel relationship with atmospheric pressure.
A parallel relationship is shown by comparing Burkholderia cepaciae phage 1 with Burkholderia cepaciae phage 781 (R2 = 0.980), and with "closely related" Bordetella phage BIP-1 (R2 = 0.561).
The early indications suggest that the Tsarnaev brothers may have had a parallel relationship.
Future studies with electron microscope are needed to characterize the subcellular localization of GAP-43 and to examine synaptophysin and its parallel relationship with GAP-43 in the dysplastic neurons.
In the original scene, the macho Miguel was trying to maintain parallel relationships with his pregnant wife and a village prostitute.
In this sense, we found that the relationships between Differential WL and DM are not moderated by expected WL (Pre-Task WL), as Differential WL and Pre-Task WL had no significant interaction and presented parallel relationship patterns with DM.
Was it a one night stand, or a long-time parallel relationship?
They asked people who had served in the military about their relationships with commanding officers, apparently in an effort to suggest a parallel with the relationship between Mr. Muhammad, a military veteran, and the younger man.
The phases were found to have a cube-on-cube (or parallel) orientation relationship with α1 L12 interfaces that were parallel to {1 0 0}.
